scholarly journals Dominating Number on Icosahedral-Hexagonal Network

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Miroslava Mihajlov Carević
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Dominating Number ◽  
Hexagonal Network

In this paper, we deal with the dominating set and the domination number on an icosahedral-hexagonal network. We will consider all cases of successive halving of the edges of triangles that are the sides of icosahedrons and thus obtain icosahedral-hexagonal networks.

scholarly journals On Triangular Secure Domination Number

2020 ◽  
Vol 2 (2) ◽  
pp. 105-110
Author(s):  
Emily L Casinillo ◽  
Leomarich F Casinillo ◽  
Jorge S Valenzona ◽  
Divina L Valenzona
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Triangular Grid ◽  
Combinatorial Formula ◽  
Grid Graph ◽  
Minimum Cardinality ◽  
Dominating Number ◽  
Triangular Number ◽  
Perfect Numbers ◽  
Secure Domination

Let T_m=(V(T_m), E(T_m)) be a triangular grid graph of m ϵ N level. The order of graph T_m is called a triangular number. A subset T of V(T_m) is a dominating set of T_m  if for all u_V(T_m)\T, there exists vϵT such that uv ϵ E(T_m), that is, N[T]=V(T_m).  A dominating set T of V(T_m) is a secure dominating set of T_m if for each u ϵ V(T_m)\T, there exists v ϵ T such that uv ϵ E(T_m) and the set (T\{u})ꓴ{v} is a dominating set of T_m. The minimum cardinality of a secure dominating set of T_m, denoted by γ_s(T_m)  is called a secure domination number of graph T_m. A secure dominating number  γ_s(T_m) of graph T_m is a triangular secure domination number if γ_s(T_m) is a triangular number. In this paper, a combinatorial formula for triangular secure domination number of graph T_m was constructed. Furthermore, the said number was evaluated in relation to perfect numbers.

scholarly journals The domination number of round digraphs

2020 ◽  
Vol 18 (1) ◽  
pp. 1625-1634
Author(s):  
Xinhong Zhang ◽  
Caijuan Xue ◽  
Ruijuan Li
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Dominating Number

Abstract The concept of the domination number plays an important role in both theory and applications of digraphs. Let D = ( V , A ) D=(V,A) be a digraph. A vertex subset T ⊆ V ( D ) T\subseteq V(D) is called a dominating set of D, if there is a vertex t ∈ T t\in T such that t v ∈ A ( D ) tv\in A(D) for every vertex v ∈ V ( D ) \ T v\in V(D)\backslash T . The dominating number of D is the cardinality of a smallest dominating set of D, denoted by γ ( D ) \gamma (D) . In this paper, the domination number of round digraphs is characterized completely.

scholarly journals Computing the split domination number of grid graphs

2021 ◽  
Vol 5 (1) ◽  
pp. 1
Author(s):  
V. R. Girish ◽  
P. Usha
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
New Method ◽  
Minimum Cardinality ◽  
Grid Graphs ◽  
Star Graphs ◽  
Dominating Number ◽  
Vertex Set ◽  
Induced Graph

<p>A set <em>D</em> - <em>V</em> is a dominating set of <em>G</em> if every vertex in <em>V - D</em> is adjacent to some vertex in <em>D</em>. The dominating number γ(<em>G</em>) of <em>G</em> is the minimum cardinality of a dominating set <em>D</em>. A dominating set <em>D</em> of a graph <em>G</em> = (<em>V;E</em>) is a split dominating set if the induced graph (<em>V</em> - <em>D</em>) is disconnected. The split domination number γ<em><sub>s</sub></em>(<em>G</em>) is the minimum cardinality of a split domination set. In this paper we have introduced a new method to obtain the split domination number of grid graphs by partitioning the vertex set in terms of star graphs and also we have<br />obtained the exact values of γ<em>s</em>(<em>G<sub>m;n</sub></em>); <em>m</em> ≤ <em>n</em>; <em>m,n</em> ≤ 24:</p>

scholarly journals Geodetic Global Domination in the Join of Two Graphs

2020 ◽  
Vol 8 (5) ◽  
pp. 4579-4583
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Dominating Number ◽  
Geodetic Set ◽  
Global Domination Number

A set S of vertices in a connected graph is called a geodetic set if every vertex not in lies on a shortest path between two vertices from . A set of vertices in is called a dominating set of if every vertex not in has at least one neighbor in . A set is called a geodetic global dominating set of if is both geodetic and global dominating set of . The geodetic global dominating number is the minimum cardinality of a geodetic global dominating set in . In this paper we determine the geodetic global domination number of the join of two graphs.

scholarly journals Hop Domination in Graphs-II

2015 ◽  
Vol 23 (2) ◽  
pp. 187-199
Author(s):  
C. Natarajan ◽  
S.K. Ayyaswamy
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Minimum Cardinality ◽  
Unicyclic Graphs ◽  
The Family ◽  
Independent Domination ◽  
Domination In Graphs ◽  
Domination Numbers

Abstract Let G = (V;E) be a graph. A set S ⊂ V (G) is a hop dominating set of G if for every v ∈ V - S, there exists u ∈ S such that d(u; v) = 2. The minimum cardinality of a hop dominating set of G is called a hop domination number of G and is denoted by γh(G). In this paper we characterize the family of trees and unicyclic graphs for which γh(G) = γt(G) and γh(G) = γc(G) where γt(G) and γc(G) are the total domination and connected domination numbers of G respectively. We then present the strong equality of hop domination and hop independent domination numbers for trees. Hop domination numbers of shadow graph and mycielskian graph of graph are also discussed.

scholarly journals Bipartite graphs with close domination and k-domination numbers

2020 ◽  
Vol 18 (1) ◽  
pp. 873-885
Author(s):  
Gülnaz Boruzanlı Ekinci ◽  
Csilla Bujtás
Keyword(s):  
Positive Integer ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
The Given

Abstract Let k be a positive integer and let G be a graph with vertex set V(G) . A subset D\subseteq V(G) is a k -dominating set if every vertex outside D is adjacent to at least k vertices in D . The k -domination number {\gamma }_{k}(G) is the minimum cardinality of a k -dominating set in G . For any graph G , we know that {\gamma }_{k}(G)\ge \gamma (G)+k-2 where \text{&#x0394;}(G)\ge k\ge 2 and this bound is sharp for every k\ge 2 . In this paper, we characterize bipartite graphs satisfying the equality for k\ge 3 and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when k=3 . We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.

A note on improved upper bounds on the transversal number of hypergraphs

2019 ◽  
Vol 11 (01) ◽  
pp. 1950004
Author(s):  
Michael A. Henning ◽  
Nader Jafari Rad
Keyword(s):  
Upper Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Size ◽  
Total Domination ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Isolated Vertex ◽  
Transversal Number

A subset [Formula: see text] of vertices in a hypergraph [Formula: see text] is a transversal if [Formula: see text] has a nonempty intersection with every edge of [Formula: see text]. The transversal number of [Formula: see text] is the minimum size of a transversal in [Formula: see text]. A subset [Formula: see text] of vertices in a graph [Formula: see text] with no isolated vertex, is a total dominating set if every vertex of [Formula: see text] is adjacent to a vertex of [Formula: see text]. The minimum cardinality of a total dominating set in [Formula: see text] is the total domination number of [Formula: see text]. In this paper, we obtain a new (improved) probabilistic upper bound for the transversal number of a hypergraph, and a new (improved) probabilistic upper bound for the total domination number of a graph.

scholarly journals The Split Distance 2 Domination in Graphs

2018 ◽  
Vol 7 (4.10) ◽  
pp. 589
Author(s):  
A. Lakshmi ◽  
K. Ameenal Bibi ◽  
R. Jothilakshmi
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Domination In Graphs

A distance - 2 dominating set D V of a graph G is a split distance - 2 dominating set if the induced sub graph <V-D> is disconnected. The split distance - 2 domination number is the minimum cardinality of a split distance - 2 dominating set. In this paper, we defined the notion of split distance - 2 domination in graph. We got many bounds on distance - 2 split domination number. Exact values of this new parameter are obtained for some standard graphs. Nordhaus - Gaddum type results are also obtained for this new parameter.  

scholarly journals New Bounds on the Double Total Domination Number of Graphs

2021 ◽  
Author(s):  
A. Cabrera-Martínez ◽  
F. A. Hernández-Mira
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Dominating Sets ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set

AbstractLet G be a graph of minimum degree at least two. A set $$D\subseteq V(G)$$ D ⊆ V ( G ) is said to be a double total dominating set of G if $$|N(v)\cap D|\ge 2$$ | N ( v ) ∩ D | ≥ 2 for every vertex $$v\in V(G)$$ v ∈ V ( G ) . The minimum cardinality among all double total dominating sets of G is the double total domination number of G. In this article, we continue with the study of this parameter. In particular, we provide new bounds on the double total domination number in terms of other domination parameters. Some of our results are tight bounds that improve some well-known results.

scholarly journals Split Domination Vertex Critical and Edge Critical Graphs

2020 ◽  
Vol 26 (1) ◽  
pp. 55-63
Author(s):  
Girish V R ◽  
Usha P
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Critical Graphs ◽  
Critical Graph ◽  
Induced Graph ◽  
Vertex Removal ◽  
Edge Addition

A dominating set D of a graph G = (V;E) is a split dominating set ifthe induced graph hV 􀀀 Di is disconnected. The split domination number s(G)is the minimum cardinality of a split domination set. A graph G is called vertexsplit domination critical if s(G􀀀v) s(G) for every vertex v 2 G. A graph G iscalled edge split domination critical if s(G + e) s(G) for every edge e in G. Inthis paper, whether for some standard graphs are split domination vertex critical ornot are investigated and then characterized 2- ns-critical and 3- ns-critical graphswith respect to the diameter of a graph G with vertex removal. Further, it is shownthat there is no existence of s-critical graph for edge addition.

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